Reflection complexity periodicity conjecture

Let $\mathbf{x}$ be a sequence. The reflection complexity periodicity conjecture.

$$r_{\mathbf{x}}(n)=r_{\mathbf{x}}(n+2)\quad\text{for some }n$$

if and only if $\mathbf{x}$ is eventually periodic. One direction is known: if $\mathbf{x}$ is eventually periodic, then the equality holds for all sufficiently large $n$. The conjecture asks whether the converse holds as well.